**Crouching AGM, hidden modularity**

Shaun Cooper, Jesús Guillera, Armin Straub, Wadim Zudilin — Preprint — 2016

## Abstract

Special arithmetic series \(f(x)=\sum_{n=0}^\infty c_nx^n\), whose coefficients \(c_n\) are normally given as certain binomial sums, satisfy `self-replicating' functional identities. For example, the equation $$ \frac1{(1+4z)^2}\,f\biggl(\frac z{(1+4z)^3}\biggr) =\frac1{(1+2z)^2}\,f\biggl(\frac{z^2}{(1+2z)^3}\biggr) $$ generates a modular form \(f(x)\) of weight 2 and level 7, when a related modular parametrization \(x=x(\tau)\) is properly chosen. In this note we investigate the potential of describing modular forms by such self-replicating equations as well as applications of the equations that do not make use of the modularity. In particular, we outline a new recipe of generating AGM-type algorithms for computing \(\pi\) and other related constants. Finally, we indicate some possibilities to extend the functional equations to a two-variable setting.## Download

Link | Size | Description | Hits |
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crouching-agm.pdf | 304.75 KB | Preprint (PDF, 16 pages) | 255 |

## BibTeX

@article{crouching-agm-2016, author = {Shaun Cooper and Jes\'us Guillera and Armin Straub and Wadim Zudilin}, title = {Crouching {AGM}, hidden modularity}, journal = {Preprint}, year = {2016}, }